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Answer

`x = (4-5k)/(7), y = (13k-9)/(7), z =k`

Explanation

The given system of equations
`3x-y +4z =3`
x +2y-3z = -2
`6x+5y +lambdaz =-3`
has atleast one solution, if `Delta ne 0`.
`therefore Delta = |{:(3, -1, 4), (1, 2, -3), (6, 5, lambda):}| ne 0`
`rArr 3(2lambda + 15) +1(lambda + 18) + 4 (5-12) ne 0`
`rArr 7(lambda + 5) ne 0`
`rArr lambda ne -5`
Let z= -k, then equations become
3x-y =3-4k
and x +2y -3k =-2
On solving, we get
`x = (4-5k)/(7), y =(13k-9)/(7), z =h`